đề thi học sinh giỏi toán

Mathematical Olympiads 2001–2002 Problems and Solutions From Around the World Copyright Information Mathematical Olympiads 2001–2002 Problems and Solutions From Around the World Edited by Titu Andreescu, Zuming Feng, George Lee, Jr., and Po-Ru Loh Published and distributed by The Mathematical Association of America MAA PROBLEM BOOKS SERIES INFORMATION 2001 National Contests: Problems and Solutions Belarus 1.1 Belarus Problem The problem committee of a mathematical olympiad prepares some variants of the contest Each variant contains problems, chosen from a shortlist of n problems, and any two variants have at most one problem in common (a) If n = 14, determine the largest possible number of variants the problem committee can prepare (b) Find the smallest value of n such that it is possible to prepare ten variants of the contest Solution: (a) The problem committee can prepare 14 variants, and no more We prove that given a shortlist of n problems, the committee · n4 variants Consider any one of the prepares at most n−1 n problems, and suppose that k variants contain that problem The other 3k problems in these variants are distinct from each other and from the chosen problem, implying that 3k ≤ n − and k ≤ n−1 Now, summing the number of variants containing each problem over the n possible problems, we obtain a maximum count of n−1 · n problems in all the variants combined Because each variant has problems, there are at most n−1 · n4 variants In particular, when n = 14, the problem committee can prepare at most 14 · 14 = 14 variants We now show that this is indeed possible Label the problems 1, , 14, with labels taken modulo 14 Then consider the following fourteen variants for t = 0, 1, , 13: {1 + t, + t, + t, + t} Take any pair A, B of distinct problems It suffices to show that the pair A, B appears in at most one variant; i.e., that there is at most one way to write (A, B) ≡ (a + t, b + t) (mod 14) with a, b ∈ {1, 2, 5, 7} and ≤ t ≤ 13 Consider the 12 pairs (a, b) with a, b ∈ {1, 2, 5, 7} and a = b The differences b − a take on 12 distinct values ±1, ±2, , ±6 over these 12 pairs Thus, there is at most one pair of values a, b ∈ {1, 2, 5, 7} with b − a ≡ B − A (mod 14) With this pair, there is at most one value t ∈ {0, 1, , 13} with A ≡ a + t (mod 14) 2001 National Contests: Problems and Solutions This completes the proof (b) Using the result in part (a), if n ≤ 12, then there are at most n−1 12 · n4 ≤ 11 3 · = variants Hence, n ≥ 13 Indeed, n = 13 problems suffice: take the 14 variants described in part (a) and remove the four variants that contain problem 14 We are left with 10 variants, as required Problem Let x1 , x2 , and x3 be real numbers in [−1, 1], and let y1 , y2 , and y3 be real numbers in [0, 1) Find the maximum possible value of the expression − x1 − x2 − x3 · · − x2 y3 − x3 y1 − x1 y2 Solution: The maximum possible value of the expression is We first rewrite the expression as follows: − x1 − x2 − x3 · · − x1 y2 − x2 y3 − x3 y1 (∗) Under the given restraints, the three numerators are nonnegative and the three denominators are positive Thus, the three fractions in the above product are nonnegative By the given inequalities, x1 (2y2 − 1) ≤ |x1 ||2y2 − 1| ≤ 1, or − x1 ≤ 2(1 − x1 y2 ) Dividing by − x1 y2 (which is positive under the given restraints), 1−x1 ≤ Applying similar reasoning shows that all we find that 1−x y2 three fractions in (∗) are at most Therefore, the three fractions in (∗) are between and 2, implying that their product is at most When x1 = x2 = x3 = −1 and y1 = y2 = y3 = 0, this bound is attained Problem Let ABCD be a convex quadrilateral circumscribed about a circle Lines AB and DC intersect at E, and B and C lie on AE and DE, respectively; lines DA and CB intersect at F, and A and B lie on DF and CF , respectively Let I1 , I2 , and I3 be the incenters of triangles AF B, BEC, and ABC, respectively Line I1 I3 intersects lines EA and ED at K and L, respectively, and line I2 I3 Belarus intersects lines F C and F D at M and N, respectively Prove that EK = EL if and only if F M = F N Solution: Let I be the incenter of quadrilateral ABCD Observe that EK = EL if and only if line KL is perpendicular to the internal angle bisector of angle AED Line KL is the same as line I1 I3 , and the internal angle bisector of angle AED is line II2 Thus, EK = EL if and only if I1 I3 ⊥ II2 Likewise, F M = F N if and only if I2 I3 ⊥ II1 Hence, it suffices to show that I1 I3 ⊥ II2 if and only if I2 I3 ⊥ II1 Observe that lines II3 and I1 I2 are the angle bisectors of the pair of vertical angles formed at B Hence, II3 ⊥ I1 I2 Thus, if I1 I3 ⊥ II2 , then I3 is the orthocenter of triangle II1 I2 , implying that I2 I3 ⊥ II1 Likewise, if I2 I3 ⊥ II1 , then I1 I3 ⊥ II2 Problem On the Cartesian coordinate plane, the graph of the parabola y = x2 is drawn Three distinct points A, B, and C are marked on the graph with A lying between B and C Point N is marked on BC so that AN is parallel to the y-axis Let K1 and K2 be the areas of triangles ABN and ACN, respectively Express AN in terms of K1 and K2 √ Solution: We will show that AN = 4K1 K2 Let A = (a, a2 ), B = (b, b2 ), and C = (c, c2 ) Without loss of generality, assume that b < c It is easy to verify that the point (a, (a − b)(b + c) + b2 ) = (a, ab + ca − bc) = (a, (a − c)(b + c) + c2 ) is on BC, implying that this point is N Thus, AN = ab + ca − bc − a2 = (a − b)(c − a) Also, we have K1 = 12 AN (a − b) and K2 = 12 AN (c − a) Thus, a − b = 2K1 /AN and c − a = 2K2 /AN Combining this with our above result gives AN = (a − b)(c − a) = or AN = √ 4K1 K2 2K1 2K2 · , AN AN 2001 National Contests: Problems and Solutions Problem Prove that for every positive integer n and every positive real a, an + 1 − ≥ n2 a + − an a Solution: By the AM-GM inequality, we know that xn−k + xn−k ≥2 when < k < n and x > If n is even, we sum the inequalities for k = 1, 3, , n − to obtain n ≥ · = n xn−1 When n is odd, we add one to the sum of the inequalities for k = 1, 3, , n − to obtain xn−1 + xn−3 + · · · + xn−1 + xn−3 + · · · + x2 + + n−1 + · · · + n−1 ≥ · + = n x x In either case, we have xn − 1/xn = xn−1 + xn−3 + · · · + n−1 ≥ n x − 1/x x In particular, setting x = a1/2 yields an/2 − 1/an/2 ≥ n a1/2 − 1/a1/2 Squaring both sides and rearranging, we obtain an/2 − an/2 ≥ n2 a1/2 − a1/2 , which, when expanded, is exactly the desired inequality Problem Three distinct points A, B, and N are marked on the line , with B lying between A and N For an arbitrary angle α ∈ (0, π2 ), points C and D are marked in the plane on the same side of such that N, C, and D are collinear; ∠N AD = ∠N BC = α; and A, B, C, and D are concyclic Find the locus of the intersection points of the diagonals of ABCD as α varies between and π2 Solution: Let R be the point between A and B satisfying AR/RB = AN/N B The locus is the circle ω with diameter N R, with N and R removed We first show that given points C, D satisfying the conditions, the intersection P of the diagonals of quadrilateral ABCD lies on ω Belarus Because quadrilateral ABCD is a trapezoid (with AD BC) and cyclic, it is an isosceles trapezoid symmetric about the perpendicular bisector of AD and BC By symmetry, N and P lie on this line, and line N P is the internal angle bisector of angle BP C (and the external angle bisector of angle BP A) Draw the line through P parallel to BC and perpendicular to P N , and let it intersect AB at R Because line P R is perpendicular to line P N (the external angle bisector of angle BP A), line P R must be the internal angle bisector of angle BP A By the Internal and External Angle Bisector Theorems, we have AR AP AN = = , RB PB NB implying that R = R Because ∠N P R = ∠N P R = π/2, P lies on the circle with diameter N R, as claimed (Alternatively, it is easy to show that line P R is the image of N under the polar transformation through circle ABCD Hence, A, R , B, N are a harmonic range and AR /R B = AN/N B.) It remains to show that any point P on ω \ {N, R} is in the locus This is simple: given such a P , reflect AB across line N P to form segment CD; C, D satisfy the required conditions Let P be the intersection of diagonals AC and BD Then P, P lie on the same line through N , and ∠N P R = ∠N P R = π/2, implying that P = P Problem In the increasing sequence of positive integers a1 , a2 , , the number ak is said to be funny if it can be represented as the sum of some other terms (not necessarily distinct) of the sequence (a) Prove that all but finitely terms of the sequence are funny (b) Does the result in (a) always hold if the terms of the sequence can be any positive rational numbers? Solution: (a) Without loss of generality, is the greatest integer that divides ak for all k (Otherwise, if d > divides every term of the sequence, then dividing each term by d does not change the problem.) Let A > be some term in the sequence, and let p1 , , pn be the primes that divide A For each k = 1, 2, , n, we can find a term Ak in the sequence such that pk | Ak We claim that all 2002 National Contests: Problems 27 for all n > Problem Let ABC be a triangle Prove that 3A 3B 3C A−B B−C C −A sin + sin + sin ≤ cos + cos + cos 2 2 2 Problem Let n be an integer greater than 2, and P1 , P2 , · · · , Pn distinct points in the plane Let S denote the union of the segments P1 P2 , P2 P3 , , Pn−1 Pn Determine whether it is always possible to find points A and B in S such that P1 Pn AB (segment AB can lie on line P1 Pn ) and P1 Pn = kAB, where (1) k = 2.5; (2) k = Problem Let n be a positive integer and let S be a set of 2n + elements Let f be a function from the set of two-element subsets of S to {0, , 2n−1 − 1} Assume that for any elements x, y, z of S, one of f ({x, y}), f ({y, z}), f ({z, x}) is equal to the sum of the other two Show that there exist a, b, c in S such that f ({a, b}), f ({b, c}), f ({c, a}) are all equal to Problem 10 Consider the family of non-isosceles triangles ABC satisfying the property AC + BC = 2AB Points M and D lie on side AB such that AM = BM and ∠ACD = ∠BCD Point E is in the plane such that D is the incenter of triangle CEM Prove that exactly one of the ratios CE EM MC , , EM MC CE is constant (i.e., is the same for all triangles in the family) Problem 11 Find in explicit form all ordered pairs of positive integers (m, n) such that mn − divides m2 + n2 28 3.14 Vietnam Vietnam Problem Let ABC be a triangle such that angle BCA is acute Let the perpendicular bisector of side BC intersect the rays that trisect angle BAC at K and L, so that ∠BAK = ∠KAL = ∠LAC = ∠BAC Also let M be the midpoint of side BC, and let N be the foot of the perpendicular from A to line BC Find all such triangles ABC for which AB = KL = 2M N Problem A positive integer is written on a board Two players alternate performing the following operation until appears on the board: the current player erases the existing number N from the board and replaces it with either N − or N/3 Whoever writes the number on the board first wins Determine who has the winning strategy when the initial number equals (a) 120, (b) (32002 −1)/2, and (c) (32002 + 1)/2 Problem The positive integer m has a prime divisor greater than √ 2m + Find the smallest positive integer M such that there exists a finite set T of distinct positive integers satisfying: (i) m and M are the least and greatest elements, respectively, in T , and (ii) the product of all the numbers in T is a perfect square Problem On an n × 2n rectangular grid of squares (n ≥ 2) are marked n2 of the 2n2 squares Prove that for each k = 2, 3, , n/2 + 1, there exists k rows of the board and k!(n − 2k + 2) (n − k + 1)(n − k + 2) · · · (n − 1) columns, such that the intersection of each chosen row and each chosen column is a marked square Problem that Find all polynomials p(x) with integer coefficients such q(x) = (x2 + 6x + 10)(p(x))2 − is the square of a polynomial with integer coefficients Problem Prove that there exists an integer m ≥ 2002 and m distinct positive integers a1 , a2 , , am such that m m a2i i=1 a2i −4 i=1 2002 National Contests: Problems is a perfect square 29 2002 Regional Contests: Problems 30 31 2002 Regional Contests: Problems 4.1 Asian Pacific Mathematics Olympia d Problem Let a1 , a2 , , an be a sequence of non-negative integers, where n is a positive integer Let a1 + a2 + · · · + an An = n Prove that n a1 !a2 ! · · · an ! ≥ ( An !) , and determine when equality holds (Here, An denotes the greatest integer less than or equal to An , a! = × × · · · × a for a ≥ 1, and 0! = 1.) Problem Find all positive integers a and b such that a2 + b b2 − a and b2 + a a2 − b are both integers Problem Let ABC be an equilateral triangle Let P be a point on side AC and let Q be a point on side AB so that both triangles ABP and ACQ are acute Let R be the orthocenter of triangle ABP and let S be the orthocenter of triangle ACQ Let T be the intersection of segments BP and CQ Find all possible values of ∠CBP and ∠BCQ such that triangle T RS is equilateral Problem Let x, y, z be positive numbers such that 1 + + = x y z Show that √ √ √ √ √ √ √ x + yz + y + zx + z + xy ≥ xyz + x + y + z Problem Find all functions f : R → R with the following properties: (i) there are only finitely many s in R such that f (s) = 0, and (ii) f (x4 + y) = x3 f (x) + f (f (y)) for all x, y ∈ R 32 4.2 Austrian-Polish Mathematics Olympiad Austrian-Polish Mathematics Olym piad Problem Let A = {2, 7, 11, 13} A polynomial f with integer coefficients has the property that for each integer n, there exists p ∈ A such that p | f (n) Prove that there exists p ∈ A such that p | f (n) for all integers n Problem The diagonals of a convex quadrilateral ABCD intersect at the point E Let triangle ABE have circumcenter U and orthocenter H Similarly, let triangle CDE have circumcenter V and orthocenter K Prove that E lies on line U K if and only if it lies on line V H Problem Find all functions f : Z+ → R such that f (x + 22) = f (x) and f (x2 y) = (f (x))2 f (y) for all positive integers x and y Problem Determine the number of real solutions of the system x1 = cos xn , x2 = cos x1 , , xn = cos xn−1 Problem For every real number x, let F (x) be the family of real sequences a1 , a2 , satisfying the recursion an+1 = x − an for n ≥ The family F (x) has minimal period p if (i) each sequence in F (x) is periodic with period p, and (ii) for each < q < p, some sequence in F (x) is not periodic with period q Prove or disprove the following claim: for each positive integer P , there exists a real number x such that the family F (x) has minimal period p > P 2002 Regional Contests: Problems 4.3 33 Balkan Mathematical Olympiad Problem Let A1 , A2 , , An (n ≥ 4) be points in the plane such that no three of them are collinear Some pairs of distinct points among A1 , A2 , , An are connected by line segments, such that every point is directly connected to at least three others Prove that from among these points can be chosen an even number of distinct points X1 , X2 , , X2k (k ≥ 1) such that Xi is directly connected to Xi+1 for i = 1, 2, , 2k (Here, we write X2k+1 = X1 ) Problem The sequence a1 , a2 , is defined by the initial conditions a1 = 20, a2 = 30 and the recursion an+2 = 3an+1 − an for n ≥ Find all positive integers n for which + 5an an+1 is a perfect square Problem Two circles with different radii intersect at two points A and B The common tangents of these circles are segments M N and ST , where M, S lie on one circle while N, T lie on the other Prove that the orthocenters of triangles AM N , AST , BM N , and BST are the vertices of a rectangle Problem Find all functions f : Z+ → Z+ such that 2n + 2001 ≤ f (f (n)) + f (n) ≤ 2n + 2003 for all positive integers n 34 4.4 Baltic Team Contest Baltic Team Contest Problem A spider is sitting on a cube A fly lands on the cube, hoping to maximize the length of the shortest path to the spider along the surface of the cube Can the fly guarantee doing so by choosing the point directly opposite the spider (i.e., the point that is the reflection of the spider’s position across the cube’s center)? Problem Find all nonnegative integers m such that (22m+1 )2 + is divisible by at most two different primes Problem Show that the sequence 2002 2003 2004 , , , , 2002 2002 2002 considered modulo 2002, is periodic Problem Find all integers n > such that any prime divisor of n6 − is a divisor of (n3 − 1)(n2 − 1) Problem Let n be a positive integer Prove that the equation 1 x + y + + = 3n x y does not have solutions in positive rational numbers Problem Does there exist an infinite, non-constant arithmetic progressions, each term of which is of the form ab where a and b are positive integers with b ≥ 2? 35 2002 Regional Contests: Problems 4.5 Czech-Polish-Slovak Mathematical C ompetition Problem Let a and b be distinct real numbers, and let k and m be positive integers with k + m = n ≥ 3, k ≤ 2m, and m ≤ 2k We consider n-tuples (x1 , x2 , , xn ) with the following properties: (i) k of the xi are equal to a, and in particular x1 = a; (ii) m of the xi are equal to b, and in particular xn = b; (iii) no three consecutive terms of x1 , x2 , , xn are equal Determine all possible values of the sum xn x1 x2 + x1 x2 x3 + · · · + xn−1 xn x1 Problem Given is a triangle ABC with side lengths BC = a ≤ CA = b ≤ AB = c and area S Let P be a variable point inside triangle ABC, and let D, E, F be the intersections of rays AP, BP, CP with the opposite sides of the triangle Determine (as a function of a, b, c, and S) the greatest number u and the least number v such that u ≤ P D + P E + P F ≤ v for all such P Problem Let n be a given positive integer, and let S = {1, 2, , n} How many functions f : S → S are there such that x + f (f (f (f (x)))) = n + for all x ∈ S? Problem Let n, p be integers such that n > and p is a prime If n | (p − 1) and p | (n3 − 1), show that 4p − is a perfect square Problem In acute triangle ABC with circumcenter O, points P and Q lie on sides AC and BC, respectively Suppose that BC AP = PQ AB and BQ AC = PQ AB Show that O, P , Q, and C are concyclic Problem Let n ≥ be a fixed even integer polynomials of the form We consider xn + an−1 xn−1 + · · · + a1 x + with real coefficients, having at least one real root Determine the least possible value of the sum a21 + · · · + a2n−1 36 4.6 Mediterranean Mathematical Competition Mediterranean Mathematical Comp etition Problem Find all positive integers x, y such that y | (x2 + 1) and x2 | (y + 1) Problem Let x, y, a be real numbers such that x + y = x3 + y = x5 + y = a Determine all positive values of a Problem Let ABC be an acute triangle Let M and N be points on the interiors of sides AC and BC, respectively, and let K be the midpoint of segment M N The circumcircles of triangles CAN and BCM meet at C and at a second point D Prove that line CD passes through the circumcircle of triangle ABC if and only if the perpendicular bisector of segment AB passes through K 2002 Regional Contests: Problems 4.7 37 St Petersburg City Mathematical Olympiad (Russia) Problem Positive numbers a, b, c, d, x, y, and z satisfy a + x = b + y = c + z = Prove that 1 + + ≥ (abc + xyz) ay bz cx Problem Let ABCD be a convex quadrilateral such that ∠ABC = 90◦ , AC = CD, and ∠BCA = ∠ACD Let E be the midpoint of segment AD, and L be the intersection point of segments BF and AC Prove that BC = CL Problem integer: One can perform the following operations on a positive (i) raise it to any positive integer power; (ii) cut out the last two digits of the integer, multiply the obtained two-digit number by 3, and add it to the number formed by the remaining digits of the initial integer (For example, from 3456789 one can get 34567 + · 89.) Is it possible to obtain 82 from 81 by using operations (i) and (ii)? Problem Points M and N are marked on diagonals AC and AM BD of cyclic quadrilateral ABCD Given that BM DN = CM and ∠BAD = ∠BM C, prove that ∠AN B = ∠ADC Problem A country consists of no fewer than 100,000 cities, where 2001 paths are outgoing from each city Each path connects two cities, and every pair of cities is connected by no more than one path The government decides to close some of the paths (at least one but not all) so that the number of paths outgoing from each city is the same Is this always possible? Problem Let ABC be a triangle and let I be the center of its incircle ω The circle Γ passes through I and is tangent to AB and AC at points X and Y , respectively Prove that segment XY is tangent to ω Problem Several 1×3 rectangles and 100 L-shaped figures formed by three unit squares (“corners”) are situated on a grid plane It is known that these figures can be shifted parallel to themselves so that 38 St Petersburg City Mathematical Olympiad (Russia) the resulting figure is a rectangle A student Olya can translate 96 corners to form 48 × rectangles Prove that the remaining four corners can be translated to form two additional × rectangles Problem The sequence {an } is given by the following relation: an+1 = (an − 1)/2 if an ≥ 1, 2an /(1 − an ) if an < Given that a0 is a positive integer, an = for each n = 1, 2, , 2001, and a2002 = 2, find a0 Problem There are two 2-pan balances in a zoo for weighing animals An elephant is on a pan of the first balance and a camel is on a pan of the second balance The weights of both animals are whole numbers, and their total does not exceed 2000 A set of weights, totaling 2000, have been delivered to the zoo, where each weight is a whole number It turns out that no matter what the elephant’s and the camel’s weights are, one can distribute some of the weights over the balances’ four pans so that both balances are in equilibrium Find the minimum number of weights that could have been delivered to the zoo Problem 10 The integer N = a0a0 a0b0c0c0 c0, where the digits a and c are written 1001 times each, is divisible by 37 Prove that b = a + c Problem 11 Let ABCD be a trapezoid such that the length of lateral side AB equals the sum of the lengths of bases AD and BC Prove that the bisectors of angles A and B meet at a point on side CD Problem 12 Can the sum of the pairwise distances between the vertices of a 25-vertex tree be equal to 1225? Problem 13 The integers from to 10 are written on a blackboard Each minute, Kolya erases three or four of the smallest integers and writes down seven or eight consecutive integers following the largest integer on the board Prove that the sum of all the integers on the blackboard is never a power of Problem 14 Find the maximal value of α > for which any set of eleven real numbers, = a1 ≤ a2 ≤ · · · ≤ a11 = 1, 2002 Regional Contests: Problems 39 can be split into two disjoint subsets with the following property: the arithmetic mean of the numbers in the first subset differs from the arithmetic mean of the numbers in the second subset by at most α Problem 15 Let O be the circumcenter of acute scalene triangle ABC, C1 be the point symmetric to C with respect to O, D be the midpoint of side AB, and K be the circumcenter of triangle ODC1 Prove that point O divides into two equal halves the segment of line OK that lies inside angle ACB Problem 16 Polygon P has the following two properties: (i) no three vertices of P are collinear; and (ii) there are at least two ways that P can be dissected into triangles by drawing non-intersecting diagonals of P Prove that some four vertices of P form a convex quadrilateral lying entirely inside P Problem 17 Let p be a prime number Given that the equation pk + pl + pm = n2 has an integer solution, prove that p + is divisible by Problem 18 An alchemist has 50 different substances He can convert any 49 substances taken in equal quantities into the remaining substance without changing the total mass Prove that, after a finite number of manipulations, the alchemist can obtain the same amount of each of the 50 substances Problem 19 Let ABCD be a cyclic quadrilateral Points X and Y are marked on sides AB and BC such that quadrilateral XBY D is a parallelogram Points M and N are the midpoints of diagonals AC and BD, and lines AC and XY meet at point L Prove that points M, N, L, and D are concyclic Problem 20 Two players play the following game There are 64 vertices on the plane at the beginning On each turn, the first player picks any two vertices that not yet have an edge between them and connects them with an edge, and the second player introduces a direction on this edge The second player wins if the graph obtained after 1959 turns is connected; otherwise the first player wins Which player has a winning strategy? 40 St Petersburg City Mathematical Olympiad (Russia) Problem 21 The shape of a lakeside is a convex centrally-symmetric 100-gon A1 A2 A100 with center of symmetry O There is a polygonal island B1 B2 B100 in the lake whose vertices Bi are the midpoints of the segments OAi , i = 1, 2, , 100 There is a jail on the island surrounded with a high fence along its perimeter Two security guards are situated at the opposite points on the lakeside Prove that every point on the lakeside can be observed by at least one of the guards Problem 22 Each of the FBI’s safes has a secret code that is a positive integer between and 1700, inclusive Two spies learn the codes of two different safes and decide to exchange their information Coordinating beforehand, they meet at the shore of a river near a pile of 26 rocks The first spy throws several rocks into the water, then the second, then the first, and so on until all the rocks are used The spies leave after that, without having said a word to each other How could the information have been transmitted? Problem 23 A flea jumps along integer points on the real line, starting from the origin The length of each its jumps is During each jump, the flea sings one of (p − 1)/2 songs, where p is an odd prime Consider all of the flea’s musical paths from the origin back to the origin consisting of no more than p − jumps Prove that the number of such paths is divisible by p Problem 24 Let ABCD be a circumscribed quadrilateral with O the center of its inscribed circle A line passes through O and meets sides AB and CD at point X and Y , respectively Given that ∠AXY = ∠DY X, prove that AX/BX = CY /DY Problem 25 Let an = Fnn , where Fn is the nth Fibonacci number (F1 = F2 = 1, Fn+1 = Fn + Fn−1 ) Is the sequence bn = √ a1 + a2 + · · · + an bounded above? Problem 26 Let a and b be positive integers such that 2a + and 2b + are relatively prime Find all possible values of gcd(22a+1 + 22a+1 + 1, 22b+1 + 22b+1 + 1) Problem 27 Let O be the center of the incircle ω of triangle ABC Let the tangency points of ω with BC, CA, and AB be A1 , B1 , and C1 , respectively The perpendicular to line AA1 at A1 meets line 41 2002 Regional Contests: Problems B1 C1 at X Prove that line BC passes through the midpoint of segment AX Problem 28 A positive integer is written on a blackboard Dima and Sasha play the following game Dima calls some positive integer x, and Sasha adds ±x to the number on the blackboard They repeat this procedure many times Dima’s goal is to get a nonnegative power of a particular positive integer k on the board Find all possible values of k for which Dima will be able to this regardless of the initial number written on the board Problem 29 Find all continuous functions f : (0, ∞) → (0, ∞) such that for all positive real numbers x and y, f (x)f (y) = f (xy) + f x y [...]... repeating the second operation k times We want to do this, however, without making any other entries in the column negative or changing an entry that is already 2 into another number Thus, whenever this is in danger of happening—that is, whenever we have an entry equal to 2—we multiply the entry’s row by 2 before performing the subtraction on the column In this way, each entry that is already equal to 2... ai+m−1 , , ai+m for i = 1, 2, , m After this, the sequence is m, m − 1, m − 2, , 1; n, n − 1, n − 2, , m + 1 Next swap block a1 , , am with am+1 , , an The total number of moves is m + 1, as desired If n is odd, then write n = 2m + 1, and for the first m moves, swap block ai , , ai+m−1 with ai+m , ai+m+1 for i = 1, 2, , m After this, the sequence is m + 1, m, m − 1, , 2;... that interval is possible If, on the other hand, we allow x, y to be negative, then XY is at a minimum when Z, X, Y lie on a line in that order This actually 20 Bulgaria happens when (x, y, z) = √ −1 2 2 3 √ ,√ , √ 21 21 7 , and t = XY 2 /3 = 12 /3 = 1/3 From this situation we can slide X, Y , and Z continuously√so that x and y increase while z decreases √ until (x, y, z) = (0, (−1+ 5)/2, 1), where we... any value √in the interval [1/3, (3 − 5)/2] It can also take any value in [(3 − 5)/2, 1], as shown above When X, Z, Y lie on a line in that order, XY reaches a maximum; this happens when (x, y, z) = (−1, −2, 2), and t = 3 here From this situation we can slide X, Y , and Z continuously √ so that √ x and y increase while z decreases until (x, y, z) = (−1/ 3, −2/ 3, 0), where t = 1 as before Therefore... Nancy is given a matrix in which each entry is a positive integer She is permitted to make either of the following two moves: (i) select a row and multiply each entry in this row by n; (ii) select a column and subtract n from each entry in this column Find all possible values of n for which given any matrix, it is possible for Nancy to perform a finite sequence of moves to obtain a matrix in which each... cases: 1 D lies on segment AC and I is the intersection between g and tB Actually, this is not possible because the intersection between g and tB is on the opposite side of line BC as point D, but A is supposed to be on the same side of line BC as point D 2 D lies on segment AC and I is X, the intersection between g and tC This is not possible Note that the angle φ between tC and g is equal to the angle... the ratio of [A B C D ] and [ABCD], obeying 4 − a2 ≤ CD, DA ≤ a We claim that the minimal choice of D is the midpoint of arc AC on the opposite side of AC as B This choice of √D clearly satisfies the latter restriction, because CD = DA = 2 in this case To show that it is indeed minimal, we decompose the areas [A B C D ] and [ABCD] into [A ACB ] + [D ACC ] and [ABC]+[ADC], respectively The first summand... onto line AC, D C ≥ AC with equality when D C AC This occurs when DO ⊥ AC, so we see that our choice of D minimizes [D ACC ] and hence [A B C D ] On the other hand, our choice of D clearly maximizes the length of the altitude from D 27 2001 National Contests: Problems and Solutions to AC, and hence the area [ADC] Thus, it maximizes [ABCD] It follows that this D minimizes [A B C D ]/[ABCD], as claimed... and ∠CXB = ∠XBJ < 90◦ , so X is too far away from C to be the incenter of any triangle with B and C as two of its vertices 3 D does not lie on segment AC and I is Ξ, the intersection between γ and τB This is not possible Let K be the intersection of lines DJ and γ Note that CK > CJ because in triangle CJK, angle C is right and angle J is equal to ∠JCD + ∠CDJ, which is greater than π/4 On the other... CX, and as X is too far away from C to be the incenter of any triangle with B and C as two of its vertices, Ξ is also too far 4 D does not lie on segment AC and I is the intersection between γ and τD This is possible Let A be the point on ray DC such that DA = DB Let the incircle of triangle A BC have center I and touch A B at P Let b = CD, c = BD, d = BC, and e = A B Note that A C = c − b Say the ... property: the triangle formed by the vertices the magpies originally sit at, and the triangle formed by the vertices they return to after the hunter passes by, are of the same type Solution: The property... proved the claim for n = k; we show that the statement is true for the case n = k + Call the squares on the edge of the (2k + 1) × (2k + 1) board boundary squares and call the rest of the squares... a−1 Because the left hand side is none other than the product of i=1 ζ i b−1 and j=1 ζ j , one of these two factors must equal Assume that it is the former; the other case is similar Then a is